### Global integrability of field theories

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Particular topics include singularity formation and the longtime behavior of solutions of nonlinear evolution equations.

## Dr Gandalf Lechner - People - Cardiff University

Differential geometry research at TU Bobenko , Pinkall , Sullivan , Suris and FU Polthier is concerned with global differential geometry of surfaces, geometric optimization problems, and the theory of integrable systems. Because it emphasizes techniques of discrete differential geometry, including applications to mathematical visualization, this research is described in more detail as part of Area 5. This entails a very rich fabric of interwoven mathematics, including such diverse topics as the differential geometry and topology of curved space-times, the representation theory of ordinary and affine non-compact super Lie algebras and Yangians, and the theory of classical and quantum integrability.

The group of Professor Kreimer Structure of local quantum field theory focuses on mathematical aspects of local, renormalizable quantum field theories as they appear in physics.

## Negative energy densities in integrable quantum field theories at one-particle level

While the physics of renormalizable field theory is the most precisely tested part of physics available, it also is a source of mathematical problems relating to algebra, algebraic geometry and number theory. Almost complex structures in 6D with non-degenerate Nijenhuis tensors and large symmetry groups.

Kruglikov, Boris; Matveev, Vladimir. The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta.

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- Negative energy densities in integrable quantum field theories at one-particle level!

Nonlinearity ; Volum 29 6. Dunajski, Maciej; Ferapontov, Eugene V. On the Einstein-Weyl and conformal self-duality equations.

### Research student

Journal of Mathematical Physics ; Volum 56 8. Letters in Mathematical Physics ; Volum Submaximally symmetric CR-structures. Journal of Geometric Analysis ; Volum Lobachevskii Journal of Mathematics ; Volum 36 3. Submaximal metric projective and metric affine structures. Differential geometry and its applications Print ; Volum Alekseevsky, D.

## Towards an operator-algebraic construction of integrable global gauge theories

Homogeneous almost complex structures in dimension 6 with semi-simple isotropy. Annals of Global Analysis and Geometry ; Volum 46 4. Symmetries of filtered structures via filtered Lie equations. Doubrov, Boris; Kruglikov, Boris.

source url On the models of submaximal symmetric rank 2 distributions in 5D. Ferapontov, Eugene V. Dispersionless integrable systems in 3D and Einstein-Weyl geometry. Journal of differential geometry ; Volum 97 2. Symmetries of almost complex structures and pseudoholomorphic foliations. International Journal of Mathematics ; Volum 25 8. Generalized Lie-Backlund theorem for Lie class overdetermined systems.

Journal of Nonlinear Mathematical Physics ; Volum 19 2. Nonexistence of an integral of the 6th degree in momenta for the Zipoy-Voorhees metric. Physical Review D. Particles and fields ; Volum 85 SDiff 2 and uniqueness of the Plebanski equation. Journal of Mathematical Physics ; Volum 53 8. The gap phenomenon in the dimension study of finite type systems. Central European Journal of Mathematics ; Volum 10 5. Finite-dimensionality in Tanaka theory. Annales de l'Institut Henri Poincare. Journal of Nonlinear Mathematical Physics ; Volum 18 4.

Anderson, Ian; Kruglikov, Boris. Rank 2 distributions of Monge equations: Symmetries, equivalences, extensions. Advances in Mathematics ; Volum 3. Involutivity of field equations. Journal of Mathematical Physics ; Volum 51 3. Compatibility of Gauss maps with metrics. Differential geometry and its applications Print ; Volum 28 2. Differential invariants of the motion group actions. Nova Science Publishers, Inc. Kruglikov, Boris Serafimovich. Invariant characterization of Liouville metrics and polynomial integrals.

Tangent and normal bundles in almost complex geometry. Kruglikov, Boris Serafimovich; Lychagin, Valentin.

Spencer delta-cohomology, restrictions, characteristics and involutive symbolic PDEs. Transactions of the American Mathematical Society ; Volum Invariants of pseudogruop actions: Homological methods and Finiteness theorem. A piece-wise affine contracting map with positive entropy.

Discrete and Continuous Dynamical Systems ; Volum Dynamics and entropy in the Zhang model of self-organized criticality. Journal of statistical physics ; Volum Entropy via multiplicity. Multi-brackets of differential operators and compatibility of PDE systems. Comptes rendus. Mathematique ; Volum Ergodic Theory and Dynamical Systems ; Volum Vanishing of the entropy pseudonorm for certain integrable systems. Electronic research announcements of the American Mathematical Society ; Volum Note on two compatibility criteria: Jacobi-Mayer bracket vs. Lobachevskii Journal of Mathematics ; Volum Kruglikov, Boris Serafimovich; Lychagina, Olga.

Finite dimensional dynamics for Kolmogorov-Petrovsky-Piskunov equation. Characteristic distributions on 4-dimensional almost complex manifolds. Banach Center Publications ; Volum Deformation of big pseudoholomorphic disks and application to the Hanh pseudonorm. Mathematique ; Volum 4. Transitive and transversal actions of pseudogroups on submanifolds. Matematika ; Volum Non-existence of higher dimensional pseudoholomorphic submanifolds.

### Deliverables

Manuscripta mathematica ; Volum Mayer brackets and solvability of PDEs. Differential geometry and its applications Print ; Volum 17 Examples of integrable sub-Riemannian geodesic flows. Journal of dynamical and control systems ; Volum 8 3. Kruglikov, Boris Serafimovich; Overholt, Marius. Pseudo-holomorphic mappings and Kobayashi hyperbolicity. Differential geometry and its applications Print ; Volum 11 3. Exact classification of divergence-free nondivergent vector fields on surfaces of small genus.

Mathematical Notes ; Volum 65 3. Existence of close pseudoholomorphic disks for almost complex manifolds and their application to the Kobayashi-Royden pseudonorm. Functional analysis and its applications ; Volum 33 1. Supergravity Padova Italy , September Gravity - New perspectives from strings and higher dimensions Benasque Spain , 14—26 July Strings Brussels Belgium , July Pre-Strings Leuven Belgium , July String Math Uppsala Sweden , July Gauge theories, supergravity and superstrings Benasque Spain , June Quantum Spacetime '19 Bratislava Slovakia.

Iberian Strings Barcelona Spain , January Quantum fields, knots, and strings Warsaw Poland , September. Vistas over the Swampland Madrid Spain , September Moonshine Vienna Austria , September Higgs bundles in mathematics and physics summer school Hamburg Germany , September Supersymmetric theories, dualities and deformations Bern Switzerland , July StringPheno Warsaw Poland , July Strings Okinawa Japan , June Holography Brussels Belgium , May Planck Bonn Germany , May Field equations on Lorentzian manifolds Hamburg Germany , March Beyond the standard model Bad Honnef Germany , March Quantum Spacetime '18 Sofia Bulgaria , February Iberian Strings Granada Spain , January Post-inflationary string cosmology Bologna Italy , September Flat connections in physics and geometry Heidelberg Germany , July Gravity - New perspectives from strings and higher dimensions Benasque Spain , July